How to know if a tangent bundle is trivial from its defining equations

Solution 1:

If a smooth $n$-manifold $M \subset \mathbb{R}^m$ is globally defined as the preimage of a regular value of a smooth function $f:\mathbb{R}^n \to \mathbb{R}^{m-n}$, then we have the desired setup $$TM=\{(x,v) \in \mathbb{R}^{m} \times \mathbb{R}^{m} \mid f(x)=0 \text{ and } df_{x}(v)=0\}.$$

If you read his answer more carefully, @jef808 was saying that we could try to use the defining equations to find linearly independent global sections. For example, one obvious solution to the equation $ux+vy=0$ is $(u,v)=(-y,x)$. Then the map $S^1\to TS^1$ given by $(x,y) \mapsto ((x,y),(-y,x))$ defines a nonvanishing vector field. We can generalize this construction to $S^{2n-1}$ and its tangent bundle \begin{align*} TS^{2n-1} &= \bigg \{((x_1,y_1,\ldots,x_n,y_n),(u_1,v_1,\ldots,u_n,v_n)) \in \mathbb{R}^{2n} \times \mathbb{R}^{2n} \mid \\ & \qquad \qquad \sum_{k=1}^n x_k^2 +\sum_{k=1}^n y_k^2=1 \text{ and } \sum_{k=1}^n u_k x_k + \sum_{k=1}^n v_k y_k =0 \bigg \} \end{align*} by defining a map $$(x_1,y_1,\ldots,x_n,y_n)\mapsto ((x_1,y_1,\ldots,x_n,y_n),(-y_1,x_1,\ldots,-y_n,x_n)).$$ However, for $n >1$, we need several linearly independent nonvanishing sections. As we found out circa 1960, this is only possible for $S^1$, $S^3$, and $S^7$; try googling "parallelizability of spheres". So even when the defining equations are very simple, it may be practically impossible to see whether or not you can construct enough linearly independent global sections of the tangent bundle. Similarly, if there was another way to use the defining equations to demonstrate triviality of a bundle, then Milnor, Bott, et cetera probably would have used it in the case of spheres.

One can ask if there are special cases where the defining equations shed light on the issue. I don't know if there are interesting families of special cases, but there are certainly ad hoc methods on a case-by-case basis:

Example (Torus). Even though we know that the torus has a trivial tangent bundle, it's not so apparent from a particular embedding $M \subset \mathbb{R}^3$, say $$M=\{(x,y,z) \in \mathbb{R}^3 \mid f(x,y,z)=(x^2 +y^2+z^2+3)^2-16(x^2+y^2)=0\}.$$ We know that points $((x,y,z),(u,v,w))$ in the tangent space satisfy \begin{align*} df_{(x,y,z)}(u,v,w)&=\begin{bmatrix} -32x +4x(3+x^2+y^2+z^2) \\ -32y +4y(3+x^2+y^2+z^2) \\ 4z(3+x^2+y^2+z^2) \end{bmatrix} \begin{bmatrix} u \\ v \\ w \end{bmatrix} \\ &=-16(2x u+2yv)+2(3+x^2+y^2+z^2)(2xu+2yv+2zw) \\ &=0.\end{align*} Similar to the $S^1$ case, we have an obvious solution of $(u,v,w)=(-y,x,0)$. Call this first vector field $\nu$. To find a second (linearly independent) vector field $\eta$, we could start by assuming $\eta=(u,v,w)$ is perpendicular to $\nu=(-y,x,0)$, i.e. $\eta=c_1(x,y,0)+c_2(0,0,1)$ for some $c_1,c_2$ (that are functions of $x,y,z$). Plugging $u=c_1 x$, $v=c_1 y$, $w=c_2$ into the above equation, we must have $$-16(2x^2 c_1+2y^2 c_1)+2(3+x^2+y^2+z^2)(2x^2 c_1+2y^2c_1+2z c_2) =0,$$ equivalently written as \begin{align*}2x^2 c_1+2y^2c_1+2z c_2 &=\frac{16(2x^2 c_1+2y^2 c_1)}{2(3+x^2+y^2+z^2)} \\ c_2 &=\frac{c_1}{z} \left(\frac{8(x^2 +y^2 )}{3+x^2+y^2+z^2} -x^2 +y^2\right). \end{align*} If we set $c_1=z$, this gives us a reasonably simple expression for $c_2$. One can easily check that $\nu(x,y,z)=(-y,x,0)$ and $\eta(x,y,z)=(x z, y z, c_2(x,y,z))$ define two linearly independent nonvanishing vector fields on $M$. Thus $M$ has a trivial tangent bundle.

Solution 2:

Since a rank $k$ vector bundle $E \to M$ is trivial iff it admits $k$ global sections which are everywhere linearly independent (in the fibers), one way to do this would be to try and write those sections explicitly from the defining equations. Of course this is a lot easier if you already know that your bundle is trivial. For example with your description of $TS^1$ the global section $\theta \mapsto (\cos \theta, \sin \theta, -\sin \theta, \cos \theta)$ shows that $TS^1$ is trivial.

In principle, having those equations for the presentation of the bundle should help finding trivializing sections. On the other hand, showing that some bundle is not trivial from this crude presentation looks rather hard since we'd lack geometric intuition...

Solution 3:

I agree with your comment:

Being able to parametrize the manifold itself would prove that the tangent bundle is trivial. Not being able to find a parametrization for a global section does not prove that the tangent bundle is non-trivial.

And I think this is essentially "how valuable" a parameterization can be for this problem (at least most of the time). A lot of great technology has been developed (e.g. Stiefel-Whitney classes) in order to answer this question, so I doubt there are direct ways of approaching it with a parametrization. I suggest Milnor-Stasheff: Characteristic Classes as a great reference which develops techniques for answering the question of whether a bundle is trivial early.